Pointwise universal trigonometric series

A series $S_a=\sum\limits_{n=-\infty}^\infty a_nz^n$ is called a {\it pointwise universal trigonometric series} if for any $f\in C(\T)$, there exists a strictly increasing sequence $\{n_k\}_{k\in\N}$ of positive integers such that $\sum\limits_{j=-n_k}^{n_k} a_jz^j$ converges to $f(z)$ pointwise on $\T$. We find growth conditions on coefficients allowing and forbidding the existence of a pointwise universal trigonometric series. For instance, if $|a_n|=O(\e^{\,|n|\ln^{-1-\epsilon}\!|n|})$ as $|n|\to\infty$ for some $\epsilon>0$, then the series $S_a$ can not be pointwise universal. On the other hand, there exists a pointwise universal trigonometric series $S_a$ with $|a_n|=O(\e^{\,|n|\ln^{-1}\!|n|})$ as $|n|\to\infty$.

Limitations of instantaneous water quality sampling in surface-water catchments: Comparison with near-continuous phosphorus time-series data.

The validity of load estimates from intermittent, instantaneous grab sampling is dependent on adequate spatial coverage by monitoring networks and a sampling frequency that re?ects the variability in the system under study. Catchments with a ?ashy hydrology due to surface runoff pose a particular challenge as intense short duration rainfall events may account for a signi?cant portion of the total diffuse transfer of pollution from soil to water in any hydrological year. This can also be exacerbated by the presence of strong background pollution signals from point sources during low flows. In this paper, a range of sampling methodologies and load estimation techniques are applied to phosphorus data from such a surface water dominated river system, instrumented at three sub-catchments (ranging from 3 to 5 km2 in area) with near-continuous monitoring stations. Systematic and Monte Carlo approaches were applied to simulate grab sampling using multiple strategies and to calculate an estimated load, Le based on established load estimation methods. Comparison with the actual load, Lt, revealed signi?cant average underestimation, of up to 60%, and high variability for all feasible sampling approaches. Further analysis of the time series provides an insight into these observations; revealing peak frequencies and power-law scaling in the distributions of P concentration, discharge and load associated with surface runoff and background transfers. Results indicate that only near-continuous monitoring that re?ects the rapid temporal changes in these river systems is adequate for comparative monitoring and evaluation purposes. While the implications of this analysis may be more tenable to small scale ?ashy systems, this represents an appropriate scale in terms of evaluating catchment mitigation strategies such as agri-environmental policies for managing diffuse P transfers in complex landscapes.

Nonlinear time series prediction based on a power-law noise model

In this paper we investigate the influence of a power-law noise model, also called noise, on the performance of a feed-forward neural network used to predict time series. We introduce an optimization procedure that optimizes the parameters the neural networks by maximizing the likelihood function based on the power-law model. We show that our optimization procedure minimizes the mean squared leading to an optimal prediction. Further, we present numerical results applying method to time series from the logistic map and the annual number of sunspots demonstrate that a power-law noise model gives better results than a Gaussian model.

NHS Inquiries: A Time Series Analysis

This article provides a time series analysis of NHS public inquiries and inquiries related to health against the background of recent policy changes which are centralizing hazardous incident investigations within agencies such as the Healthcare Commission.

Natural history of idiopathic erythrocytosis - A retrospective case series review

Idiopathic Erythrocytosis (IE) is a diagnosis given to patients who have an absolute erythrocytosis (red cell mass more than 25% above their mean normal predicted value) but who do not have a known form of primary or secondary erythrocytosis (BCSH guideline, 2005). We report here the results of a follow-up study of 80 patients (44 male and 36 female) diagnosed with IE from the United Kingdom and the Republic of Ireland over a 10 year period. Baseline information was initially collected when investigating for molecular causes of erythrocytosis in this group. The diagnosis of IE was made on the basis of a raised red cell mass >25% above mean normal predicted value, absence of Polycythaemia Vera (PV) based on the criteria of Pearson and Messinezy (1996), and the exclusion of secondary erythrocytosis (oxygen saturation >92% on pulse oximetry, no history of sleep apnoea, no renal or hepatic pathology, and a normal oxygen dissociation curve (if indicated). The average age at diagnosis of erythrocytosis was 34.5 (2–74 years). Erythropoietin levels were available for 77/80 of the patients and were low in 18 (23%) and normal or high in 59 (74%). Ultrasound imaging was carried out in 67 patients (84%) at time of diagnosis and no significant abnormalities found. Fourteen patients had a family history of erythrocytosis. These patients have now been followed up for an average of 9.4 years (range 1–39). Out of 80 patients 56 patients can still be classified as having IE, of whom 52 are living (cause of death in the other 4 - lung cancer, RTA, sepsis, unknown). Thirty-five of these patients are regularly venesected, 3 take hydroxyurea (one also venesected), 11 receive no treatment while treatment is unknown in 2. Twenty take aspirin, 1 warfarin and 31 no thromboprophylaxis. Four of these patients had suffered thromboembolic complications (3 with CVA/TIAs and 1 with recurrent DVT) at or before their original diagnosis. Since diagnosis 8 patients have had 9 thrombotic events of which 7 were arterial (1 CVA, 3 TIAs, 1 MI, 2 PVD) and 2 venous (DVT/PE). Twenty take aspirin, 1 dipyridamole, 1 warfarin and 30 take no thromboprophylaxis. Out of the 24 patients who now have a diagnosis other than IE, 8 have been diagnosed with myelo-proliferative disease. Thirteen patients have a molecular abnormality which is likely to account for their erythrocytosis (11 VHL, 1 PHD-2, 1 EPO-receptor mutations). Three patients have secondary erythrocytosis. Older case studies identified a heterogenous group of patients, some of whom probably had apparent erythrocytosis and some who had either primary polycythaemia or secondary causes later identified (Modan and Modan, Najean et al). More recent reviews have identified a more homogenous group with low rates of transformation to myelofibrosis/acute leukaemia and low rates of thrombosis of around 1% patient-year. Follow up of our initial patient group does indeed reveal a heterogeneous group of patients with 10% now diagnosed with an MPD, although when analysis is confined to those patients who continue to fulfil the criteria for IE, the clinical course has been more stable. There has been no progression to MDS or leukaemia in this group (one patient with PV progressed to AML). The rate of thrombosis is 1.6% patient-years which is lower than the rate seen in PV and is consistent with the rate identified in other series. Molecular defects continue to be identified in this group and future investigation is likely to reveal further abnormalities.